Inverse Sub‐structuring Method for Rigidly Coupled Product Transport System based on Frequency Response Function Testing Probe Technique

The inverse sub‐structuring method has been recently proposed and applied for inverse analysis of product transport system, to predict the component‐level frequency response functions (FRFs) and the coupling dynamic stiffness from only the system‐level FRFs. However, previous applications of this method were all developed based on the assumption that the components were coupled by flexible couplings. Actually, increasing more components are welded or bolted to construct a coupled system, which should be treated as rigidly coupled system. The aim of this paper is to derive a new FRF‐based inverse sub‐structuring method for the analysis of the dynamic characteristics of a two‐component coupled product transport system with rigid couplings. And then a so‐called FRF testing probe technique is proposed and applied to measure the difficult‐to‐monitor FRFs at the coupling interface. The developed method is verified by a lumped‐mass model, showing exact agreement. Finally, the experiment on a physical prototype of two‐substructure coupled product transport system is performed to further check the accuracy of the suggested method. The proposed method is an extension of previous inverse sub‐structuring method and may help to obtain the main controlling factors and contributions from the various structure‐borne paths for product transport system. Copyright © 2016 John Wiley & Sons, Ltd.     

Inverse Sub‐structuring Method for Multi‐coordinate Rigidly Coupled Product Transport System based on a Novel Shearing Probe Technique

The inverse sub‐structuring method can predict the component‐level frequency response functions (FRFs) of product (critical component) for product transport system from only measured system‐level FRFs, facilitating the cushioning packaging design. However, the FRFs of the coupling interface between product and vehicle are usually of extreme difficulty to be measured due to the limited accessible space. To overcome this difficulty, the authors suggested a so‐called FRF probe technique method in the previous study, which may be more suitable for the single‐coordinate coupled system. In practice, most of the product transport systems should be treated as multi‐coordinate coupled system. The aim of this paper is to derive a new FRF‐based inverse sub‐structuring method for multi‐coordinate rigidly coupled product transport system and develop a new shearing probe technique to obtain the difficult‐to‐monitor FRFs at the coupling interface, which will be validated by a lumped mass model and finite element models, respectively, showing perfect agreement. Finally, the experiment on a physical prototype of multi‐coordinate rigidly coupled product transport system is performed to further check the feasibility of the application prospect of the shearing probe technique for inverse analysis of product transport system. The method proposed in this study will provide the packaging designers an alternative method to monitor the integrity of product transport system. Copyright © 2017 John Wiley & Sons, Ltd.     

Indirect Inverse Substructuring Theory for Coupling Dynamic Stiffness Identification of Complex Interface Between Packaged Product and Vehicle Transport System

Inverse substructuring method has been recently proposed and applied for inverse analysis of the dynamical response of product transport system. The component‐level frequency response functions (FRFs) and the coupling dynamic stiffness for facilitating the cushioning packaging design are all predicted from only the system‐level FRFs. However, the system‐level FRFs from coupling degree of freedoms may not be measured accurately because of the difficulties of vibration excitation and response measurement for the coupled interface between packaged product and vehicle within the limited accessible space. The aim of this paper is to develop a new FRF‐based indirect inverse substructuring method for the analysis of the dynamic characteristics of a three‐substructure coupled product transport system without measuring system‐level FRFs at the coupling degree of freedoms. By enforcing the dynamic equilibrium conditions at the coupling coordinates and the displacement compatibility conditions, a closed‐form analytical solution to inverse sub‐structuring analysis of multi‐substructure coupled product transport system is derived based on the relationship of easy‐to‐monitor component‐level FRFs and the system‐level FRFs at the coupling coordinates.. The proposed method is validated by a lumped mass‐spring‐damper model, and the predicted coupling dynamic stiffness is compared with the direct computation, showing exact agreement. Then, the FRF tests of a physical prototype of multi‐substructure coupled product transport system are performed to further check the accuracy of the suggested method. The method developed offers an approach to predict the unknown coupling dynamic stiffness from measured FRFs purely. The proposed method may help to obtain the main controlling factors and contributions from the various structure‐borne paths for product transport system. Copyright © 2014 John Wiley & Sons, Ltd.     

Inverse Sub‐Structuring Method for Multi‐Coordinate Coupled Product Transport System

A new high‐accuracy transfer function is selected, and an inverse sub‐structuring method is developed for the analysis of the dynamic characteristics of a three‐sub‐structure coupled product transport system. The closed‐form analytical solution to inverse sub‐structuring analysis of multi‐coordinate coupled multi‐ sub‐structure product transport system is derived. The proposed method is validated by a lumped mass spring damper model; the predicted frequency response functions (FRFs) of sub‐structures and the coupling stiffness, in addition to the most concerned system‐level FRF, are compared with the direct computations, showing exact agreement. Then, FRF tests of a physical prototype of the multi‐coordinate coupled product transport system with three sub‐structures are performed to further check the accuracy of the suggested method. The method developed offers an approach to predict the unknown sub‐structure‐level FRFs and coupling stiffness purely from system‐level FRFs. The suggested method may help obtain the main controlling factors and contributions from the various structure‐borne paths for the product transport system, which may certainly facilitate the cushioning packaging design. Copyright © 2013 John Wiley & Sons, Ltd.     

获取界面响应不可测数据的频响探针关键技术研究

目的针对传统逆子结构理论在求解过程中界面响应难以实测的问题,提出一种利用频响探针技术来获取该界面响应的关键技术。方法首先基于该频响探针的动力学微分方程,从理论上推导了该频响探针技术的理论公式,然后对建立的二级单点刚性耦合系统进行了有限元数值验证,将利用频响探针技术预测得到的难测原点频响函数与有限元计算值进行比较,并将该预测值代入逆子结构理论公式中,得到了部件频响函数的预测值,将该预测值与有限元计算值进行了对比验证。结果预测值与有限元计算值高度吻合,验证了该理论的准确性。结论该频响探针关键技术在获取界面响应不可测数据方面,有很好的应用价值。     

Dynamic Characteristic Analysis of Refrigerator‐Truck Transport System by Using Inverse Substructure Method

This paper is a continuation of the previous research. The refrigerator‐truck system is treated as a two‐substructure multi‐coordinate coupled system, which is composed of a refrigerator substructure and a truck substructure coupled by numerous packaging units. Applying the two‐substructure multi‐coordinate coupled inverse substructure method to a product‐transport system, the paper makes an analysis of the dynamic characteristics for the refrigerator‐truck system. In order to validate the method in the refrigerator‐truck system, the measured system‐level frequency response functions (FRFs) were used to predict the substructure‐level FRFs that were compared with those directly measured and found to be in agreement in trend. To evaluate the vibration transmissibility of packaging units, the concepts of the vibration contribution, vibration contribution ratio, overall vibration contribution and overall vibration contribution ratio to product response through each packaging unit are introduced. The vibration contributions to the refrigerator through four coupling points almost coincide with each other in the whole frequency domain. However, the vibration contribution ratios are different at each frequency. The overall vibration contributions to the refrigerator through the frequency domain 10 to 50 Hz through four coupling points are respectively 9.4, 7.5, 9.8 and 11.2 mm/N, and the overall vibration contribution ratios are respectively 0.25, 0.20, 0.26 and 0.29. The vibration transmissibilities of four coupling points are similar. Copyright © 2014 John Wiley & Sons, Ltd.     

产品-包装-运载体系统动态特性研究

为缓冲防震系统的设计与评估提供一种新的结构动态分析方法,将复杂耦合结构系统的动态逆子结构分析方法应用于运输包装工程中.从系统分析的角度建立用于确定"产品-包装-运载体"在系统与部件水平的动态特性的实验技术理论.通过对一个集总参数模型实例在系统水平的传递函数的计算,验证了所建立理论的有效性.     

多部件耦合包装系统逆子结构分析一般性理论

建立多部件耦合包装系统动力学模型,建立其动力学分析的一般逆子结构方法,得到从系统水平传函反向计算各耦合部件水平传函及部件间耦合动刚度的理论公式,采用集总参数模型数值校验了所建立理论的正确性。研究结论为复杂系统动力学特性分析提供了一种新的技术方法。     

Application of the Inverse Substructure Method in the Investigation of Dynamic Characteristics of Product Transport System

Product, packaging and vehicle constitute a complex product transport system in logistics. It is very difficult to obtain accurately the dynamic response of a product transport system under the action of environmental vibration and shock. In this paper, product transport system is treated as a two substructure‐coupled system composed of product system (including critical element) and vehicle connected by packaging and its fixing (location pattern, securing, etc.); the inverse substructure method is applied to the analysis of the dynamic characteristics of the system. For verification of the validity of the inverse substructure method for product transport system, a typical lumped mass model is taken as an example for numerical validation. To check out the accuracy of the method, we completed the experiment, and the predicted substructure‐level frequency response functions are in overall agreement with those measured. The sensitivity of the method to measurement error is also made. To study the influences of the product parameters, packaging and its fixing, we investigated the effects of the coupling stiffness, mass ratio, frequency parameter ratio and damping on the dynamic response of critical element. Reducing the coupling static stiffness of product–vehicle interface can effectively lower the response of critical element, especially when the coupling stiffness is less than the stiffness of product and vehicle. Copyright © 2011 John Wiley & Sons, Ltd.     

运输包装车辆装配系统直接与间接逆子结构动态分析方法实验验证

针对"产品-包装-车辆"装配系统中的包装耦合体,基于系统与部件水平的实测频率响应函数(FRF),计算了装配耦合动刚度,包括直接逆子结构法和间接逆子结构法。设计了一套实验模型并测量、计算和分析了其系统装配耦合动刚度,实验验证了这2种方法在实际包装耦合体的设计及其缓冲防震性能分析上应用的有效性。     

一种针对含非线性子结构多点耦合运输包装系统的解耦方法

目的 考虑到运输包装系统耦合形式复杂,包装材料及包装结构具有非线性特性,不容易测量局部物理参数,需要对传统逆向子结构方法进行优化,使之能够求解非线性多点耦合系统中子结构的动态响应特性。方法 使用描述函数法将非线性的运输包装系统线性化,测量其在若干特定振动幅值下的频率响应函数;之后,应用逆向子结构方法和参数识别方法,计算包装件的模态参数;最后,拟合包装件模态参数与振动幅值之间的关系,构建函数来描述包装件的动态响应特性。结果 在集总参数模型中,解耦预测值与实际值吻合;在有限元模型中,对响应峰值的预测误差小于5%,对响应跳跃现象所在频率的预测误差小于3%。结论 该研究将传统逆向子结构方法的应用范围拓展到了非线性多点耦合系统,对复杂运输包装系统动力学模型的构建和防振包装的设计具有指导意义。     

Investigation of Frequency Response Function of Product‐Transport System Based on Multi‐coordinate Coupled Inverse Substructure Method

The product and vehicle, through a packaging, constitute a complex product‐transport system in logistics. It is very important to obtain the frequency response functions of the product‐transport system and its substructures for the design of product packaging. In this paper, the product‐transport system is treated as a two‐substructures multi‐coordinate coupled system. It is composed of a product substructure and a vehicle substructure, which are connected by a packaging structure consisting of many packaging units. The multi‐coordinate coupled inverse substructure method is developed and used to analyse the dynamic characteristics of the product‐transport system. To verify the validity of this method for the product‐transport system, the experiment of a physical prototype is conducted. The results show that the predicted substructure‐level frequency response functions are in accordance with the measured. Copyright © 2013 John Wiley & Sons, Ltd.     

基于逆子理论的洗衣机运输系统动态特性分析

目的获取产品运输包装系统各部件的动态特性。方法采用多级系统分布解耦法,结合二级刚性耦合系统逆子结构理论,推导由产品、车辆部件水平和系统水平频响传函预测关键部件频响传函的理论公式。搭建电机-洗衣机-车辆三级刚柔耦合运输系统,对理论方法进行验证。结果通过在线实验验证,基于理论预测得到的关键部件频响传函和测试值相吻合。结论研究结果为产品运输包装设计和优化提供了参考依据。     

多点耦合包装系统动态逆子结构理论试验验证

将产品-包装-运载体系统作为整体考虑,建立了二级多点动态逆子结构理论与模型。推导了用系统水平FRF来预测部件水平FRF的表达式。用弹簧-质量块组成的实际模型进行了实验验证。实验结果验证了用系统水平FRF预测部件水平FRF的可行性与准确性。     

基于试验数据的频响函数综合法概述

利用子结构方法对舰艇上复杂隔振系统进行建模分析时,因子结构的复杂性而无法准确获取其动力学模型,一种解决办法是采用试验测得的子结构频响函数数据进行基于频响函数的子结构综合。为提高子结构频响函数测试结果的精度和基于试验数据的频响函数综合结果的精度,采用频响函数综合算法、子结构频响函数测试误差的消除方法、子结构转角自由度频响函数的测试、频响函数数据的测试及影响因素等几个方面进行总结,并采用基于试验数据的频响函数综合法对一复杂隔振系统进行建模分析,得到较好的效果。可在进行实际隔振系统的子结构测试和综合时提供参考。     

多级刚性包装运输系统间接逆向子结构分析

目的针对各子部件耦合界面之间的系统水平频响函数难以测量,刚性耦合系统逆向子结构分析方法无法顺利应用的情况,提出多级系统间接分析方法。方法基于子结构理论,提出单点耦合和多点耦合系统的多级刚性耦合系统间接逆向子结构分析方法,然后建立相对应的集总参数模型,利用已知参数和公式获得部件频响函数直接计算值和预测值,最后将两者进行对比验证。结果部件频响函数直接计算值与预测值相吻合,验证了方法的准确性。结论提出的方法可为逆子结构理论在解决耦合界面频响函数难测问题时提供新思路,以及为在运输包装领域更广泛的应用提供更多的可能性。     

Elastic wave propagation in a 3‐D unbounded random heterogeneous medium coupled with a bounded medium. Application to seismic soil–structure interaction (SSSI)

We present a sub‐structuring method for the coupling between a large elastic structure, and a stratified soil half‐space exhibiting random heterogeneities over a bounded domain and impinged by incident waves. Both media are also weakly dissipative. The concept of interfaces classically used in sub‐structuring methods is extended to ‘volume interfaces’ in the proposed approach. The random dimension of the stochastic fields modelling the heterogeneities in the soil is reduced by introducing a Karhunen–Loéve expansion of these stochastic fields. The coupled overall problem is solved by Monte‐Carlo simulation techniques. A realistic example of a large industrial structure interacting with an uncertain stratified soil medium under earthquake is finally presented. This case study and others validate the presented methodology and its ability to handle complex mechanical systems. Copyright © 2002 John Wiley & Sons, Ltd.     

基于频响函数的逆子结构方法误差分析

目的 通过分析随机误差在基于试验频响函数(FRFs)的逆子结构分析方法中的传递,得出随机误差对预测结果的影响规律,为基于逆子结构方法分析复杂结构的动态特性提供参考价值.方法 对获得的系统频响函数施加不同程度(1％,5％,10％)的随机误差,对比分析各个耦合系统频响函数对预测子结构频响函数的影响.结果 对耦合系统频响函数施加随机误差后,采用逆子结构方法对耦合系统解耦后预测的子结构频响函数严重偏离真实值,尤其是共振频率附近,所施加的随机误差在预测子结构频响函数中甚至被放大了数十倍,导致预测结果不可靠;且耦合系统耦合点处的频响函数对预测结果的影响最大.结论 通过分析明确了系统频响函数所携带的随机误差对预测结果的影响规律,且这些误差将随着矩阵的求逆运算被放大,且交叉耦合系统频响函数对预测结果的影响最为显著.     

A Study of Nonlinear Effects in a Cushion‐Product System on its Vibration Transmissibility Estimates with the Reverse Multiple Input–Single Output Technique

The Reverse Multiple Input–Single Output (R‐MISO) identification technique was implemented to determine vibration transmissibility frequency response functions (FRFs) of a cushion‐product system that accounts for nonlinearities known to exist in such systems. The signed quadratic x|x| nonlinear term was used. The results were compared with the classical linear approximation FRFs calculated with the single input–single output identification method. Comparisons were made for five types of expanded polystyrene (EPS) cushions with varying density and two levels of band‐limited random excitation. It was shown that vibration transmissibility FRFs obtained with the R‐MISO method and with the inclusion of a signed quadratic x|x| nonlinear term significantly (up to ~70%) improved the total coherence of the nonlinear vibration transmissibility FRFs for the EPS cushions subjected to the static load selected to coincide with the optimum of their respective cushion curves. The R‐MISO technique was successful in accounting for some of the nonlinearities in the EPS cushion‐product system. However, its ability to deal with nonlinearities is not specific to the EPS but rather common to cushion‐product systems subjected to vertical vibrations. Copyright © 2012 John Wiley & Sons, Ltd.     

刚柔耦合包装系统动态特性分析的逆子结构方法

目的以三级刚柔耦合包装系统为研究对象,推导由系统水平传递函数反向预测部件水平传递函数的逆向子结构分析公式。方法结合刚性和柔性系统特点,分别将三级刚柔耦合系统转化为二级刚性和二级柔性结构,经过理论推导得出结果,并用集总参数模型进行验证,对理论公式进行误差分析。结果验证了理论公式的正确性,获得了各测量传函对预测传函的影响规律。结论为刚柔耦合包装系统动态力学分析提供了理论依据。